If $$f''\left( x \right) = - f\left( x \right)$$ and $$g\left( x \right) = f'\left( x \right)$$ and $$F\left( x \right) = {\left( {f\left( {\frac{x}{2}} \right)} \right)^2} + {\left( {g\left( {\frac{x}{2}} \right)} \right)^2}$$ and given that $$F\left( 5 \right) = 5,$$ then $$F\left( {10} \right)$$ is equal to :
A.
5
B.
10
C.
0
D.
15
Answer :
5
Solution :
$$\eqalign{
& F'\left( x \right) = \left[ {f\left( {\frac{x}{2}} \right).f'\left( {\frac{x}{2}} \right) + g\left( {\frac{x}{2}} \right).g'\left( {\frac{x}{2}} \right)} \right] \cr
& {\text{Here, }}g\left( x \right) = f'\left( x \right){\text{ and }}g'\left( x \right) = f''\left( x \right) = - f\left( x \right) \cr
& {\text{So, }}F'\left( x \right) = f\left( {\frac{x}{2}} \right).g\left( {\frac{x}{2}} \right) - f\left( {\frac{x}{2}} \right).g\left( {\frac{x}{2}} \right) = 0 \cr
& \Rightarrow \,F\left( x \right){\text{ is constant function}} \cr
& {\text{So, }}F\left( {10} \right) = 5 \cr} $$
Releted MCQ Question on Calculus >> Differentiability and Differentiation
Releted Question 1
There exist a function $$f\left( x \right),$$ satisfying $$f\left( 0 \right) = 1,\,f'\left( 0 \right) = - 1,\,f\left( x \right) > 0$$ for all $$x,$$ and-
A.
$$f''\left( x \right) > 0$$ for all $$x$$
B.
$$ - 1 < f''\left( x \right) < 0$$ for all $$x$$
C.
$$ - 2 \leqslant f''\left( x \right) \leqslant - 1$$ for all $$x$$
If $$f\left( a \right) = 2,\,f'\left( a \right) = 1,\,g\left( a \right) = - 1,\,g'\left( a \right) = 2,$$ then the value of $$\mathop {\lim }\limits_{x \to a} \frac{{g\left( x \right)f\left( a \right) - g\left( a \right)f\left( x \right)}}{{x - a}}$$ is-
Let $$f:R \to R$$ be a differentiable function and $$f\left( 1 \right) = 4.$$ Then the value of $$\mathop {\lim }\limits_{x \to 1} \int\limits_4^{f\left( x \right)} {\frac{{2t}}{{x - 1}}} dt$$ is-